OK, so I got trough the previous problems.
First of all I applied heat sink to the transistor to cool it down. It's much better now, but still the heat sink is a bit warmer than it should be. Maybe some little fan will be attached. Since the heat sink doesn't get hot all the time I will proceed observing this process. For transistor cooling I used old heat sink from CPU and also applied special paste that would better distribute heat between transistor and heat sink.
About the second problem (described in last blog post) - I borrowed oscilloscope to see what is happening with the pulse signal that should control the servo.
To see how the circuit works at first I didn't add the control current - power supply that produces current from 0-10V.
These are the results:
1. Potentiometer in the center position (on NE555 timer pin #5 current is 1.6V):
As you can see pulse signal is normal, nothing unusual.